How to have skeleton/armature change with shape keys?
For the eyes consider adding a vertex group for a eye [bone].  This should dominate any envelope influence if the [multiply] option is disabled.

Link to Blender documentation

https://docs.blender.org/manual/en/dev/rigging/armatures/bones/properties/deform.html#armature-bones-envelope

Multiply

This option controls how the two deforming methods interact, when they are both enabled. By default, when they are both active, all vertices belonging to at least one vertex group are only deformed through the vertex groups method